3.1 Overview of the research area and SWMM model research

The study area is located in a Xiaozhai area in the south of city X, which is a typical old urban area, with many hardened roads and large population density. At present, the waterlogging in Xiaozhai area is relatively serious. Non-point source pollution in Xiaozhai is caused by runoff pollution in the early stage of rainstorm. The cause of initial rain pollution is that there are a large number of acid gases in the atmosphere at the beginning of rainy season. These acid gases include automobile exhaust and industrial exhaust, which are wet by rain and condensed by rain ^[16,17]. After the rainwater falls to the ground, the roof pavement and surface soil layer will be washed. Under various conditions, the rainwater within 15 minutes before the rainstorm contains a lot of pollutants. To solve this problem, SWMM is introduced to simulate the runoff and pollutant accumulation scouring results in Xiaozhai area, and PSO is used to optimize based on the evaluation of the current LID facility layout scheme. Fig. 1 shows the simulation process of SWMM.

From Fig. 1, SWMM includes left and right plates. The left plate mainly carries out hydrological correlation analysis. First, it collects rainfall and underlying surface data to calculate the net rainfall, and then calculates the permeable surface runoff, impermeable surface runoff with retention, and impermeable surface runoff without retention. Secondly, the sub-basin discharge is calculated. The right plate is mainly used for routing simulation. First, the process lines of each tributary at the inlet are combined, and the next section of data is read and input into the transmission module. Secondly, the calculation mode of this section (river section or pipe section) is adjusted and the calculation is carried out to the next node. Finally, the results are output. If the output results do not exit, it will return to the left plate, and the output result exit will display the outlet flow hydrograph. In addition, SWMM in essence can be divided into hydrological, hydraulic and water quality simulation modules according to the simulation module. Hydraulic simulation is mainly aimed at the rainfall behavior of various pipelines, nodes and water storage facilities. Water quality simulation is mainly applied to water quality routing based on pollutant growth of different land types. As the specific area of the study and analysis is a large area, four LID facilities in SWMM, including sunken green space, rainwater garden, permeable pavement and green roof, are selected for layout, which are subsequently represented by A${\sim}$D.

3.2 Macro configuration optimization decision index system for sponge facilities

The layout of LID facilities has a significant effect on water quality improvement and water quantity regulation in urban areas, and it is also the core of sponge city development. Therefore, to more intuitively understand the application effect of LID facilities in the reconstruction of old urban areas, it is necessary to build a decision-making index system for macro-configuration optimization of sponge facilities. Combined with the actual situation of the old urban area of city a and considering the content of the ``sponge city construction evaluation standard'', a comprehensive index evaluation system is constructed. It includes the node overload rate, landscape effect and public acceptability in the humanistic effect. The emphasis standard includes runoff control rate, Suspended Substance (SS) load reduction rate and flood peak reduction rate. There are 8 macro-allocation decision indicators of sponge facilities, including infrastructure cost and maintenance cost in economic factors. The node overload rate in the human effect is calculated and expressed by equation (1) ^[18,19].

In equation (1), $\xi $ is the node overload rate. $\lambda _{k} $ represents the actual number of overloaded nodes. $\lambda _{0} $ means the total number of nodes. The runoff control rate in the emphasis standard is expressed by equation (2).

In equation (1), $J$ represents runoff control rate (expressed by O). $\psi $ is runoff coefficient. The SS load reduction rate is expressed by equation (3).

In equation (3), $W$ is the actual load reduction rate of pollutants (expressed by Q). $S_{j} $ is the total amount of pollutants in runoff. $S_{z} $ is the total amount of pollutants. The flood peak reduction rate is expressed by equation (4).

In equation (4), $\varsigma $ is the peak reduction rate. refers to the peak disc$R_{k} $harge after adding sponge facilities. $R_{k} $ refers to the peak discharge at the discharge port under the traditional mode development. Finally, Fig. 2 shows the constructed comprehensive index system.

From Fig. 2, the index system can obtain the optimal proportion of green facility layout after three processes, and then replace the model for inspection. Then the experiment determines whether it is the optimal layout scheme. If yes, the optimal layout scheme will be output. If not, the grey facilities will be added at a fixed point to replace the model for inspection and repeat the process. The index system displays the overall optimizing procedure of sponge facilities' layout. It takes the simulating results of SWMM as the original data and uses swarm intelligence optimization algorithm to obtain sponge green facilities' optimal proportion. For the problem area, the grey facilities are strengthened at fixed points, and by continual circulation, it can get the optimal layout scheme of sponge grey and green combination.

3.3 Multi-objective optimization configuration in old urban areas

In the MOP module in Fig. 2, the constructed mop mathematical model contains three elements: optimization objective function, decision variables and constraints. The decision scalar is determined by the objective function and constraints. In determining the objective function, the objective function obtained by using the standard deviation method is expressed by equation (5).

In equation (5), $maxE$ is the target optimization value. $\varpi _{J} $, $\varpi _{W} $ and $\varpi _{A} $ are the weights of runoff control objectives, runoff pollutant control objectives, and cost control objectives, respectively. $B$ stands for standard deviation. $A$ is the capital construction and maintenance cost (in millions) (expressed by C). The calculation of target weight mainly includes subjective weight and objective weight. Because AHP in subjective weighting can decompose the elements related to decision-making into objectives, criteria, schemes and other levels. On this basis, qualitative and quantitative analysis are carried out. The CRITIC method in the objective weighting can integrate the variation effect of indicators and the contradiction between indicators. Therefore, the AHP-CRITIC hybrid weighting method is constructed to determine the weight of the index. After obtaining the objective and subjective weights, the results of CRITIC method and AHP method are comprehensively weighted. And the subjective weights are supplemented with the calculation results of objective weights, so as to improve the accuracy of subjective weights. Since the three independent variable scenarios in the overall objective are different, it should build the correlation between facilities A${\sim}$D's layout ratios and $J$, $W$ and $A$, which is expressed by equations (6) to (8).

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In equation (6), $b_{1} $, $b_{2} $, $b_{3} $, and $b_{4} $ represent the layout proportion of facilities A${\sim}$D.

Equations (6) and (7) are nonlinear functions, so the coefficients fitted in their equations are obtained by using the response surface method.

The constraint conditions among the three elements are proposed to avoid the pursuit of infrastructure maintenance costs and ignore the control effect that should be achieved in sponge city construction, which is expressed by equations (9) and (10).

From equation (9), the layout proportion of facilities A${\sim}$D is between 0${\sim}$15%.

15% in equation (10) refers to 15% of the total area of the study area, which is mainly limited by the study area's urban nature. Based on this, the optimization of mop mathematical model is studied. Because PSO has the advantages of high precision and fast convergence compared with other algorithms, it is studied as a multi-objective optimization algorithm. It is briefly described mathematically here. Assuming that a given searchable space is a multidimensional space and there are multiple random particles, the $l$-th random particle is represented by equation (11).

In equation (11), $D_{l} $ represents the $l$-th random particle. $F$ is the dimension of searchable space. The velocity and individual extremum of the $l$-th random particle are expressed by equation (12).

In equation (12), $G_{l} $ and $\kappa _{best} $ are the velocity and individual extremum of the $l$-th random particle. Therefore, the global extremum of the whole particle swarm is represented by equation (13).

In equation (13), $\upsilon _{best} $ represents the global extremum of the particle swarm. The update speed and position of particles when searching the optimal value are expressed by equations (14) and (15).

In equation (14), $\varphi $ is the inertia weight. $h_{1} $ and $h_{2} $ are acceleration constants. $\tau _{1} $ and $\tau _{2} $ represent uniform random numbers with values between $[0$, $1]$.

In order to enhance the novelty of PSO in the multi-objective optimization analysis of old urban areas under the development of smart cities, the inertia weight of particles in the PSO algorithm can be dynamically adjusted to balance the global search and local search capabilities during the search process. In the initial stage, a larger inertia weight is used to enhance the global search ability, and gradually the weight is reduced to improve the local search accuracy. Adopting a hierarchical search strategy, the complex optimization problem is decomposed into multiple simple subproblems, each level is solved using independent PSO, and finally the final optimization solution is obtained by integrating the results of each level. Conduct sensitivity analysis on key parameters in the PSO algorithm, identify which parameters have a significant impact on algorithm performance, and design adaptive or intelligent parameter selection mechanisms based on this to enhance the robustness and adaptability of the algorithm.

4.1 Performance Analysis of PSO Algorithm

In the development of smart cities, the performance of PSO algorithms can usually be compared and analyzed through indicators such as convergence speed, final convergence accuracy, convergence stability, and execution efficiency. In order to highlight the superiority of the improved PSO algorithm, it is compared with PSO algorithm and Simulated Annealing Algorithm (SAA). Among them, simulated annealing algorithm is a global optimization algorithm based on thermodynamic principles, which has good global search ability and robustness, and performs well in solving problems such as combinatorial optimization and continuous optimization. Select a certain number of multi-objective renovation datasets in old urban areas, and compare the convergence speed, final convergence accuracy, convergence stability, and execution efficiency of the three algorithms. The results are shown in Fig. 3.

Figs. 3(a), 3(b), 3(c), and 3(d), respectively, represent the convergence speed, final convergence accuracy, convergence stability, and execution efficiency of the three algorithms. From Fig. 3, it can be seen that compared with PSO and SSA, the improved PSO has a faster convergence speed, higher final convergence accuracy, better convergence stability, and the fastest execution efficiency. These advantages make the improved PSO more competitive compared to traditional PSO algorithms, SSA, etc. in solving complex optimization problems, greatly improving the search efficiency and quality of the algorithm. At the same time, it also enhances the robustness and stability of the algorithm, indicating that it is more suitable for application in multi-objective optimization analysis of old urban areas.

4.2 Analysis of the effect of existing sponge projects

SWMM refers to the model of Zeng et al. to identify the existing problems in the sensitive areas prone to flooding in the old urban area of city A. And it has specially set relevant parameters in consideration of the actual situation of the study area ^[20]. Table 1 shows the final parameters of SWMM model.

From Table 1, the manning coefficient of impervious area and permeable area, and the storage capacity of impervious area and permeable area were 0.013, 0.15, 2, and 5, respectively. The maximum and minimum infiltration rate and attenuation coefficient of Houghton infiltration were 76 mm/h, 3.03 mm/h, and 2d${}^{-1}$, respectively, and the roughness coefficient of pipeline was 0.013. Under this parameters setting, Fig. 4 shows the effect of water quantity and quality controlling of the built sponge facilities.

From Fig. 4(a), the increase of rainfall intensity increased the surface runoff, and the surface runoff changed the most, from 2.36 mm to 53.16 mm. From Figs. 4(b) and (c), when the rainfall return period exceeded 5a, the overloaded nodes and pipe sections increased rapidly, indicating that the built LID facilities had a weak ability to cope with rainstorms. The overload rates of nodes and pipelines at 50a were 31.70% and 50.19%, respectively. From Fig. 4(d), the increase of rainfall return period gradually reduced the amount of surface growth. In general, the built sponge facilities have a certain regulation ability for water quality and quantity. However, its runoff controlling rate and SS load reducing rate under the 17.2 mm rainfall condition do not meet the requirements, and its ability to cope with rainstorms is weak.

4.3 Analysis of multiple LID facility combination scenarios

Through SWMM simulation, the built sponge facilities are difficult to cope with high-intensity rainfall and the relevant conditions of low-intensity rainfall also fail to meet the requirements. Therefore, it needs to be improved and optimized. Before the reconstruction, it is necessary to analyze the effect of LID facility scenario setting. It mainly includes single-LID facility, two-LID facility combinations, three-LID facility combinations and four-LID facility combinations. In the order of equation (6), the four single-LID facilities account for 3%, 6%, 9% and 12% of the area of the study area, while the different facility combinations do not exceed 15%. Therefore, there are 5 scenarios for each single-LID facility, a total of 20 scenarios, 60 scenarios for two facility portfolios, 40 scenarios for three facility portfolios, and 5 scenarios for four facility portfolios. The rainfall intensity of 17.2 mm and 50 A were mainly observed. Fig. 5 shows the runoff and pollutant control results of a single-LID facility.

In Fig. 5, $w$ indicates no LID facility. From Figs. 5(a) and (b), the runoff control rates under the rainfall intensity of 17.1 mm and 50a without LID facilities were 35.30% and 18.53%. The runoff control rate of facility a alone increased by 19.12%-91.35% and 27.45%-44.67%, respectively. From Figs. 5(c) and (d), the pollutant load reduction rates under the two rainfall intensities without LID facilities were 4.93% and 27.28%, respectively. The reduction rate of adding facility a alone was the highest, increasing to 16.45%-48.41% and 30.06%-79.07%. Fig. 6 shows the runoff and pollutant control results of the two facility combinations.

From Fig. 6(a), the growth values of runoff control ability of combination A+B under two rainfall intensities were the largest, which were 81.01%-90.02% and 27.91%-42.89%. From Fig. 6(b), the pollutant reduction capacities of A+B combination under two rainfall intensities were the largest, increasing to 29.95%-46.49% and 42.96%-76.05%. These results show that there is a positive relationship between water quantity control and water quality control. Fig. 6 shows the runoff and pollutant control results under the three facility combinations.

From Fig. 7(a), the runoff control capacities of combination A+B+C under the two rainfall intensities increased by 82.04%-88.04% and 30.15%-40.14%, respectively. From Fig. 7(b), the pollutant reduction capacities of A+B+C combination under two rainfall intensities were the largest, increasing to 82.04%-88.04% and 30.15%-40.14%. Fig. 8 shows the runoff and pollutant control results under the four facility combinations.

From Fig. 8(a), the runoff controlling rates of these four facilities under two rainfall intensities increased to 82.55%-85.55% and 31.86%-36.97%. Due to the small difference in the general layout proportion of LID, the net flow controlling effects had little difference, but they could control the runoff under the rainfall condition of 17.2 mm. From Fig. 8(b), the pollutant load reduction rates of the four facilities under two rainfall intensities increased to 34.87%-40.01% and 54.74%-64.94%, respectively. It is worth noting that the combination of the four facilities achieves good control effect under the condition of 17.20 mm rainfall, but still cannot meet the control effect of super rainfall.

4.4 Analysis of multi-objective renovation and optimization in old urban areas

Considering the influence of other index factors in the actual old city reconstruction project planning, the study studies the four facility combinations under the increasing proportion of facility a. The optimal layout scheme is obtained by using the mop mathematical model, and the infrastructure maintenance cost index is added. Fig. 9 shows the weights of three indicators obtained by AHP-CRITIC.

From Fig. 9(a), the subjective and objective weights were 0.25 and 0.31, respectively. From Fig. 9(b), the subjective and objective weights were 0.25 and 0.20, respectively. From figure 9(c), the subjective and objective weights were 0.50 and 0.49, respectively. On the basis of these weight results, 128 groups of schemes in Figs. 4 to 7 are input into equations (6) to (8) as independent variables, and each scheme is fitted. And regression analysis is carried out on the basis of the fitting results. Fig. 10 shows the actual regression analysis results of the probability and variance of the normal distribution of residuals.

From Fig. 10(a), the linear relationship of all scatter points was very obvious, which showed that the fitting accuracy of these curves was very high. From Fig. 10(b), the determination and correction coefficients were more than 0.7, meaning that the predicted value's reliability was high. Based on this, the weight results in equation (8) are substituted into the function obtained in equation (5) as PSO's fitness function. $F$ and $\varphi $ are 4 and 0.8, $h_{1} $ and $h_{2} $ are 0.5, the population size is 500, the accuracy is 0.001, and the maximum iterations are 300. The matrix laboratory is used to iteratively solve the optimization goal. Fig. 10 shows the simulation results of the fitness change of the optimal evolutionary individual of PSO and the optimal LID scheme.

From Fig. 11(a), the optimal deployment ratios of the final four facilities were simulated to be 3.88%, 7.95%, 2.01%, and 0.99%, respectively. From Figs. 11(b) and 11(c), the best scheme was much stronger than the non-LID facility in terms of various indicators of water quantity and quality. From Fig. 11(d), the four deployment ratios obtained by substituting the results into equations (6) to (8) were similar to the simulation results, which proved the rationality of this MOP model. On this basis, the grey facility coordination layout is carried out according to the process in Fig. 2 to make up for the control of precipitation conditions for 50a in the MOP model. Fig. 12 shows the results.

Compared with Figs. 12(a) and 12(b), the overload pipe sections and overflow nodes were significantly reduced under the rainfall condition of 50a. Gray and green facilities were reasonably connected, and green facilities played a role. During the rainfall corresponding to the design return period of rainwater pipe and channel, there was no ponding. During the rainstorm corresponding to the recurrence period of waterlogging control, no waterlogging occurred.